On some methods for the simultaneous determination of polynomial zeros
type: Item
instance of: scholarly article
title: On some methods for the simultaneous determination of polynomial zeros
zbMATH Open document ID: 0859.65047
DOI: 10.1007/BF03167248
author: Sachio Kanno
author: Nikolay Kyurkchiev
author: Tetsuro Yamamoto
published in: Japan Journal of Industrial and Applied Mathematics
publication date: 1997-04-07T00:00:00Z
review text: The authors consider some iteration methods for the simultaneous approximation of all zeros of a polynomial \(P(z)\). The methods are related to that of Weierstrass (and other authors independently) which uses the iteration formula \(\widehat z_i=z_i-P(z_i)/\prod^n_{\substack{ k=1\\ k\neq i}} (z_i-z_k)\), where the \(z_i\), \(i=1,2,\dots,n\), are approximations to the zeros of \(P(z)\). A unified convergence analysis is given for some known methods having cubic convergence rate. It is also shown that an SOR-like acceleration of the Durand-Kerner method converges for a real acceleration parameter in the range \((0,2)\). Numerical examples illustrating the theoretical results are given and include examples of solution trajectories.
Mathematics Subject Classification ID: 65H05
Mathematics Subject Classification ID: 30C15
Mathematics Subject Classification ID: 12Y05
zbMATH DE Number: 927212
zbMATH Keywords: zeros of polynomials
zbMATH Keywords: simultaneous iterative methods
zbMATH Keywords: Weierstrass method
zbMATH Keywords: numerical examples
zbMATH Keywords: convergence
zbMATH Keywords: SOR-like acceleration
zbMATH Keywords: Durand-Kerner method
reviewed by: Adrian Swift
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf03167248
OpenAlex ID: W2001799574
description: scientific article; zbMATH DE number 927212